Manuscript Oai212-A@1 · Oai212-A.tex · theorem · 79323841

Theorem: Exact realization-wise OTOC–FPP identity

Bridge 1: an exact controlled operator-spreading model › The local timer rule · lines 590–625 · proof included or linked

Oai212-A@1#thm:exact-otoc

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For Specification spec:cnot,

C_{τ} (x, t) = 1_{{T_{G} (0, x) \leq t}}, E C_{τ} (x, t) = P (T_{G} (0, x) \leq t) .

The equality holds on every positive-timer realization, with a specified tie convention if necessary.

Proof. A CNOT from $u$ to $v$ obeys

CNOT_{u \to v}^{†} Z_{v} CNOT_{u \to v} = Z_{u} Z_{v}, CNOT_{u \to v}^{†} Z_{u} CNOT_{u \to v} = Z_{u} .

Conjugate $Z_{x}$ backwards through the chronological tree circuit. If $x$ has been flagged, the only gates that enlarge its backwards $Z$ product are the parent edges on its unique ancestry chain. Consequently $Z_{x} (t)$ is the product of $Z$ over that chain, including $0$ and $x$ . If $x$ has not been flagged, it has neither been a successful target nor a successful control, and $Z_{x} (t) = Z_{x}$ .

For $x \neq = 0$ , the resulting Pauli product anticommutes with $X_{0}$ exactly when $x$ has been flagged. For $x = 0$ , it always anticommutes, matching $T_{G} (0, 0) = 0$ . A Hermitian unitary Pauli product that anticommutes with $X_{0}$ has commutator squared norm $4 I$ ; a commuting one has zero. Lemma lem:timers finishes the proof.

Displayed formulas in this passage

unlabeled · line 593
$C_{τ} (x, t) = 1_{{T_{G} (0, x) \leq t}}, E C_{τ} (x, t) = P (T_{G} (0, x) \leq t) .$
unlabeled · line 604
$CNOT_{u \to v}^{†} Z_{v} CNOT_{u \to v} = Z_{u} Z_{v}, CNOT_{u \to v}^{†} Z_{u} CNOT_{u \to v} = Z_{u} .$

Theorem: Exact realization-wise OTOC–FPP identity

Displayed formulas in this passage

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